Director

Executive office, management

Estimating extreme multivariate quantile regions

When simultaneously monitoring two possibly dependent, positive risks oneis often interested in quantile regions with very small probability p. These extreme quantile regions contain hardly or no data and therefore statistical inference is difficult. In particular when we want to protect ourselves against a calamity that has not yet occurred, we take p<1/n, withn the sample size. We consider quantile regions of the form {(x,y): x,y>0, f(x,y)<a}, where f is the joint density. Such a region has the property that it consists of the less likely points and hence that its complement is as small as possible. Using extreme value theory, we construct a natural, semiparametric estimator of such a quantile region and prove a refined form of consistency. The results could be applied in, e.g., aviation safety. As an illustration, we compute the estimated quantile regions forsimulated data sets. This is joint work with Laurens de Haan.